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2.2. Principle of nuclear magnetic relaxation in liquids 11measurements are based on the aqueous phase T 2 . During the gel reaction T 2 decreasesand levels off after several hours. The results are in agreement with the gel times obtainedfrom tilting test tube experiments. Our analysis discusses the hydrogen T 2 spectra of theaqueous phase in detail and particularly the role of methanol in the solution.In the following section we discuss a model that describes the relaxation in pure liquids.It also adequately describes the temperature and concentration dependency of T 1 for thebinary TMOS-oil mixtures. The model shows that multi-exponential relaxation behavioris expected in liquid mixtures. Furthermore, it will show that for different fluids, havingdifferent viscosities, differences in relaxation times are expected.2.2 Principle of nuclear magnetic relaxation in liquidsThis section summarizes briefly the main mechanisms of nuclear spin relaxation for hydrogennuclei in pure liquids and simple binary mixtures.2.2.1 Pure liquidsConsider a liquid which is placed and magnetized in an external magnetic field. If thelongitudinal nuclear magnetization M z is reduced to zero, for instance by applying a 90 ◦radio frequency (rf) pulse, it will relax back to its equilibrium magnitude M z (0) due tospin-lattice relaxation (see also Appendix A). The restoring magnetization is describedby an exponential[ (M z (t) = M z (0) 1 − exp − t )], (2.3)T 1where T 1 is the longitudinal relaxation time [36, 50]. The transverse magnetization M T ,which is equal to M T (0) just after the excitation of the system by the 90 ◦ pulse, decaysback to zero due to spin-spin relaxation. This is described by the exponential(M T (t) = M T (0) exp − t ), (2.4)T 2where T 2 is the transverse relaxation time [36, 50].The relaxation processes, both for T 1 and T 2 , are due to intra- and inter-molecularinteractions of the hydrogen nuclei and due to spin-rotational (SR) interactions [41, 51].The overall relaxation times T 1 and T 2 are therefore given by1T 1=1T 2=( 1T 1)( 1T 2)intraintra( ) 1+T 1( ) 1+T 2interinter( ) 1+ ,T 1 SR(2.5)( ) 1+ .T 2 SR(2.6)Intra- and inter-molecular relaxation is driven by (fluctuating) dipole-dipole interactionsbetween the hydrogen nuclei. In case of intra-molecular relaxation the interactions aremodulated by the rotational motion of the molecule. The rotational correlation time (or
12 2. Coupled mass transfer and sol-gel reaction in a two-phase bulk system”tumbling” time) τ c is of the order of 1–100 ps, which is often much shorter than theLarmor precession time of the nuclei, so that ω 0 τ c ≪ 1, where ω 0 is the Larmor precessionfrequency. This situation is referred to as the fast motion limit [50]. In this limit (T 1 ) −1intra= (T 2 ) −1intra , and the intra-molecular relaxation rate for a many-nuclei molecule is given by[52]( 1T 1,2)intra= 3 2(( µ0) 2γ 4 2 2 ∑4π n pi>j1r 6 ij)τ c , (2.7)where µ 0 , γ, , and n p are the magnetic permeability, the gyromagnetic ratio, Planck’sconstant, and the number of hydrogen nuclei per molecule, respectively. r ij are the distancesbetween the nuclei i and j. Eq. 2.7 shows that the relaxivity is proportional toa single correlation time, provided that the molecule is rigid, i.e. the distances r ij andthe orientations of the nuclei remain constant. However, internal motion or anisotropicrotation leads to multiple correlation times [53].An effective, Arrhenius type expression for the correlation time can be used assumingthat the motions are thermally activated [54]:( )τ c,eff = τ ′ EAexp , (2.8)RTwhere τ ′ , E A , R, and T are the inverse frequency factor, the activation energy for rotationalmotion, the gas constant, and the temperature, respectively.The inter-molecular contribution is linked to translational motion of the molecules.In the fast motion limit (where (T 1 ) −1inter = (T 2) −1inter ), and with the approximation that allhydrogen nuclei are located at the center of the molecule, the relaxivity is expressed as[50, 55] ( 1=T 1,2)inter π ( µ0) 2 γ 4 2 N 05 4π aD , (2.9)where N 0 is the number of hydrogen nuclei per unit volume, D is the diffusivity of themolecules and a is the closest radius of approach.Finally, we neglect spin-rotational interactions, which are only important for someliquids containing small molecules, or gaseous systems [56]. The right-hand side of Eq.2.5 therefore, reduces to the first two terms. From Eqs. 2.5, 2.7 and 2.9, the overallrelaxivity for a single-component liquid is given by1T 1,2=(( µ0) 2γ 4 2 3 ∑4π n pi>j1τrij 6 c,eff + π N 05 aD). (2.10)Although, in the fast motion limit the transverse relaxation time T 2 is equal to T 1 , theexperimentally observed relaxation time T 2 is sensitive to magnetic field inhomogeneitiesor gradients, and is often shorter than T 1 .2.2.2 Correlation of relaxation times with viscosity in pure liquidsThe relaxation processes in the liquid are driven by the molecular motions. Since theviscosity η, which is a macroscopic bulk property, is related to the molecular motions,
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158 Appendix EAs the saturations, d
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160 Bibliography[12] B. Vinot, R. S
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162 Bibliography[40] M. D. Mantle a
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164 Bibliography[70] F. J. M. van d
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166 Bibliography[97] K. Yoshida, A.
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168 Bibliography[127] G. W. Scherer
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170 Bibliography
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172 Summaryand gelation rates. The
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